Travelling Salesman Problem (TSP)
Vivek Gupta(11816)
Chetan Dalal(11218)
Karan Singh(11354)
Guide: Rizwan Hudda
(Strategist at Tower Research Capital)
Introduction
Definition:
Given a set of cities and the distance between
each possible pair, the Travelling Salesman
Problem is to find the best possible way of
‘visiting all the cities exactly once and returning
to the starting point’
A Brief History
 The problem was first defined in the 1800s by the Irish




mathematician W.R. Hamilton and the British mathematician
Thomas Kirkman.
It was, however, first formulated as a mathematical problem
only in 1930 by Karl Menger.
The name Travelling Salesman Problem was introduced by
American Hassler Whiteney.
The origin of TSP lies with Hamilton's Icosian Game, which was
a recreational puzzle based on finding a Hamiltonian cycle.
Richard M. Karp showed in 1972 that the Hamiltonian cycle
problem was NP-complete, which implies the NP-hardness of
TSP. This supplied a mathematical explanation for the apparent
computational difficulty of finding optimal tours.
TSP : An NP-HARD Problem!
 TSP is an NP-hard problem in combinatorial optimization
studied in theoretical computer science.
 In many applications, additional constraints such as limited
resources or time windows make the problem considerably
harder.
 Removing the constraint of visiting each city exactly one
time also doesn't reduce complexity of the problem.
 In spite of the computational difficulty of the problem, a
large number of heuristics and exact methods are known,
which can solve instances with thousands of cities.
Applications of TSP
• Even in its purest form the TSP, has several applications
•
•
•
•
such as planning, logistics, and the manufacture of
microchips.
Slightly modified, it appears as a sub-problem in many
areas, such as DNA sequencing.
In these applications, the concept city represents, for
example, customers, soldering points, or DNA fragments
The concept distance represents travelling times or cost, or
a similarity measure between DNA fragments.
As TSP is a NP hard problem it is often used as a
benchmark for optimization techniques.
Applications of TSP
Mechanical arm
 When a mechanical arm is used to fasten the nuts for assembling
parts, it moves through each nut in proper order and returns to the
initial position.
 The most economical travelling route will enable the mechanical arm
to finish its work within the shortest time.
Integrated circuit
 Inserting electrical elements in the manufacturing of integrated
circuits consumes certain energy when moving from one electrical
element to the other during manufacturing.
 We need to arrange the manufacturing order to minimize the energy
consumption.
In both these cases, TSP is required to be solved as a sub-problem.
Useful Variations of TSP
Variations of TSP might be desired in certain real-life scenarios
 In transportation-based graphs, the edge weights can change
constantly when they are based on the time to traverse the
roadway they represent.
 It might not be necessary to visit every node in a route, so a
route that visits only a predefined subset of nodes needs to be
determined.
 A node can be visited more than once if that provides a faster
route.
 Visiting a certain unwanted node might be useful in obtaining a
faster route.
Solving the Problem
Exact Solution
• A direct solution is to try out all the permutations and find out
•
•
•
•
•
the shortest of all these permutations.
This would be O(n!) time complexity, which implies 10^64 for 50
cities. This is practically useless.
Other approaches attempted at solving the problem include
Dynamic Programming, branch and bound, linear programming.
None of them were feasible for number of cities over 500. It has
not been determined whether an exact algorithm for TSP that
runs in O(1.9999^n) time exists.
So attempts have are made to solve TSP approximately which
give sub optimal solutions.
Here we present a few intelligent approaches to solve TSP
approximately.
An Attempt to solve TSP
using A-Star
Solving TSP using A-star
 As A-star is general graph search algorithm it can be used




for solving TSP.
As the length of the tour for the best solution is not known
the first challenge is to determine the goal of the A-Star
search.
With the given configuration one can find the upper and
lower bounds for the best solution.
Then do a binary search running A-star each time to check
if the given goal length is feasible.
Alternatively one can have terminating condition instead
of a goal and run the A-star search until the condition is
met.
A state is a path in the graph, that visits any node at most
once.
Heuristic and State Space
 At any point during the execution we have a path formed
up till that point of time and a set of vertices which have
to be added to the path to complete the tour .This forms
the state space .
 We represent a state as S(i1 ,i2..ik) which means the path
formed so far is i1 ,i2..ik and the remaining vertices are to
be added to the tour.
 A heuristic can be the weight of MST of the remaining
vertices.
How good is this State Space?
 Let S(k) denote the set of all states which have a
formed a path for k vertices.
 Number of elements in S(k) is P(n,k) = (n!/(n-k)!)
 So total number of states would be
S = P(n,0) + P(n,1) + P(n,2) …… P(n,n-1) + P(n,n)
S < n! + n! + n! ……. n! + n!
S < (n+1)n!
S < (n+1)!
 The number of states is of the order of factorial of n
But we don’t need to store the state space.
Defining the Edges
 The edge weight between two states is given as follows





X =S(i1 ,i2..ik)
Y1 =S(i0 ,i1, i2,..ik)
Wt(X,Y1) = wt(i1, i0)
Y2 =S(i1 ,i2..ik+1)
Wt(X,Y2) = wt(ik+1, ik)
Wt(A,B) is the weight of the edge A-B in the A-star state
space and wt(a,b) is the weight of the edge a-b in the given
graph.
ik and i0 should not belong the set {i1 i2 ……….ik}
No other edges exist
Thus for a given state one can find all its neighbors by
adding each of the unvisited nodes to the path.
So for each element in S(k) there are 2(n-k) neighbours.
Feasibility
 If a goal length is given we can check the feasibility of the goal in the





following way.
If there is one element of set S(n) in the open list, then the search is
performed for a fixed number of iterations (call it maxIter) from that
point of execution.
In each iteration search in the open list for the solution of desired
length.
If one such solution is found, then declare that the given length is
feasible and terminate the algorithm.
If no such solution is found within the maxIter number of iterations
then the given length is declared as not feasible.
This means that one may miss the exact solution depending upon the
value of max iterations.
Terminating condition
 If there is no fixed goal length given, then the algorithm’s
terminating condition will be as follows.
 There are n! elements in the set of solutions S(n) (not
necessarily the best)
 The approach can be running the search until K elements
of the set S(n) are in the open list .
 Once this condition is met give the best among these K
elements is declared as the solution and the algorithm is
terminated.
Genetic Algorithms
Genetic Algorithms for TSP
 Genetic algorithms are a whole new class of
Algorithms which mimic nature and evolution.
 They follow Darwin's theory of survival of the fittest.
 They are randomized algorithms and may not find
best solution.
 Using Genetic algorithms we can easily solve problems
of size less than 100.
Genetic Algorithms for TSP
 Genetic algorithm starts with a random population,
and generates modern and new population
 It replaces the weak candidates by newly formed
strong and useful candidates
 In TSP, a permutation is a candidate for population of
Genetic Algorithms (GA)
 Graph can trivially be converted to a completely
connected graph, by inserting edges having infinite
weight.
Genetic Algorithms for TSP
Genetic algorithm has many steps in it
 Evolution
 finding out good members from the population
 Cross Over
 combining two parent members to form two new different
members(children) with similar characteristics as parents.
 Mutation
 Some of newly formed members are altered randomly
 Keeps them from becoming identical and converging to a local
minima
 Completing the graph
 Missing edges are inserted with a weight of Infinity (INF)
Genetic Algorithms for TSP
 Select some random permutations of cities as initial






population
Pick some permutations randomly & reproduce in that
subset
In a selected subset, replace worst 2 permutations with 2
new generated permutations
Best 2 permutations in subset are made to be parents
Goodness of permutation is inversely related to path
length of that permutation
Smaller path length implies better permutation
Process of generating new permutation from 2 old
permutations is called cross over
Cross Over
Parent 1 - FAB | ECGD
Parent 2 - DEA | CGBF
Combining first part of parent 1 with 2nd part of
parent 2 and vice versa
Child1 – FAB CGBF
Child2 – DEA ECGD
 Children permutations are not valid permutations,
Need different crossover algorithm
Cross Over
 Instead of considering the permutation, we can only




consider the neighbors of each city in permutation
e.g., in FABCEGD, neighbors of A are F and B.
For cities have same pair of neighbors in both parent,
will have same pair of neighbors in both children also
Otherwise neighbors of a city will come as first
neighbor of that city in one parent and second in
other parent and vice versa for other children.
E.g., if neighbors of A are F, B in parent 1 and C, D in
parent 2. Then new neighbors of A are F,D and C,B
 Mutation
 To avoid generating similar kind of children, some
permutations are selected randomly, and they are
changed little bit.
 E.g., In a permutation, we can swap any two cities.
 Do we ever stop?
 After finite number of generations or
 When permutations start becoming very similar
Ant Colony Algorithm
Ant Colony Algorithm (ACA)
• This Algorithm is inspired by the social life of ants
• Individual ants are un-intelligent and they are practically
blind
• But with their social structure ants complete the complex
task of finding the shortest path without knowing
problem’s existence.
• How do they achieve this? (Next Page)
ACA - Background
• The ants have a place where they store their food (lets call
•
•
•
•
it a nest).
When they sense some food nearby their task is to carry
the food to the nest (preferably in the shortest path)
Ants leave behind a chemical substance called pheromone
which lets the other ants identify that an ant has been
there before.
The amount of pheromone that an ant deposits is inversely
proportional to the distance it has travelled.
So the ants that move along smaller paths secrete more
amount of pheromone per unit length.
ACA – The Ant’s way
• In the colony ,an ant (called blitz) moves randomly around
the colony
• If it discovers a food source, it returns directly to the nest,
leaving in its path a trail of pheromone
• These pheromones attract nearby ants which will be
inclined to follow the track
• Returning to the colony, these ants will strengthen the
route they follow.
ACA – The Ant’s way
ACA – The Ant’s way
• If there are two routes to reach the same food source then
the shorter one will be traveled by more ants than the long
route as the pheromone density is high along them
• The short route will be increasingly enhanced, and
therefore become more attractive;
• The long route will eventually disappear as pheromones
are volatile
• Eventually, all the ants take shortest route.
ACA into TSP
• Ants are simple agents which travel from one city to
•
•
•
•
another city.
Each edge has some pheromone quantity associated with
it
The pheromone quantities change dynamically with the
ants’ motion in the graph
An ant decides to take an edge depending upon the edge
length and the pheromone quantity
The pheromone in each edge is decreased by a constant
factor with time.
Pseudo Code
procedure ACO algorithm for TSPs
Set parameters, initialize pheromone trails
while (termination condition not met) do
ConstructSolutions
UpdateTrails
end
end ACO algorithm for TSPs
ACA into TSP - details
• Initially, all the ants are placed at randomly chosen cities.
• At each city, the ant probabilistically chooses a city not
yet visited , based on pheromone trail strength on the arc
and the distance between them .
• Each ant has a limited form of memory, called a Tabu list
in which the current partial tour is stored.
ACA into TSP - details
• The probability
that an ant k moves to city-j
when at city-i in iteration number t is given by the
following relation.
•
gives the list of feasible neighbors of city-i for the
ant k .
• The symbol
gives the pheromone quantity for a
given edge ij in iteration number t.
• The symbol
gives the inverse of the edge weight for
the edge between city-i and city-j.
Pheromone Update
• When all the ants have completed a solution, the trail
update is done as per the below equations
• L gives the tour length of a given ant
• One can interpret this as probabilistic version of A-star
where there is no coming back if one finds a bad path.
Performance
 ACA with some optimizations can give solutions which are




very close to the actual solution (less than 1.1 times the
actual best solution).
The number of iterations should be 5 to 10 times the
number of cities.
The number of ants should be of the order of number of
cities.
For number of cities set to 500 the algorithm gives very
good results in 4-5 minutes (around 3000 iterations).
As the algorithm is probabilistic one cannot always expect
the same performance.
Comparision of ACA and Genetic
Algorithmic Approach
Oliver30,att48,Eil51 are instances of TSP taken from TSPLIB which is a library
of sample instances for the TSP. The last column is the best solution available
in TSPLIB
ACS: Ant Colony System’s solution, GA: Basic Genetic Algorithm’s solution
 Ant colony algorithm gives good solutions very close




optimal path but the lack of pheromone in initial stages
leads to slower speed of convergence.
Genetic Algorithms have rapid global searching capability,
but the feedback of information in the system is not
utilized to its fullest extent
This may lead to redundant iterations and inefficient
solution.
As the number of cities increase ACA becomes slow GA
might not give the optimal paths.
So mixed approach is applied is being tested. Research is
going on about how to mix both approaches and make it
faster and give optimal paths.
Conclusion
 Evolutionary algorithms like ACA, Genetic Algorithms
are very successful in solving the TSP.
 These techniques with some variation can be applied
to most of the NP-hard problems in general NP-hard
problems can be modeled as graph problems.
 As we have seen in the case of ACA and Genetic
Algorithms naturally occurring phenomena can be
effectively used to model computer algorithms.
 So observe nature. Nature has solution to everything!
References
 Russel S. and Norvig P., "Artificial Intelligence: a Modern




Approach", Prentice Hall, (1998)
Adaptive Ant Colony Optimization for the Traveling
Salesman Problem, Michael Maur (2009 )
An Ant Colony Optimization Algorithm for the Stable
Roommates Problem Glen Upton(2002)
The Advantage of Intelligent Algorithms for TSP, Traveling
Salesman Problem, Theory and Applications, Prof. Donald
Davendra and Yuan-bin Mo (2010).
Comparative Analysis of Genetic Algorithm and Ant Colony
Algorithm on Solving Traveling Salesman Problem,
Kangshun Li, Lanlan Kang, Wensheng Zhang, Bing Li(2007)
References
 http://comopt.ifi.uni-heidelberg.de/software/TSPLIB95/
 http://www.lalena.com/AI/Tsp/
 http://blogs.mathworks.com/pick/2011/10/14/traveling-
salesman-problem-genetic-algorithm/
 http://en.wikipedia.org/wiki/Travelling_salesman_problem